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A class of implicit symmetric symplectic and exponentially fitted Runge–Kutta–Nyström methods for solving oscillatory problems
 Huai Yuan Zhai^{1},
 Wen Juan Zhai^{2} and
 Bing Zhen Chen^{3}Email authorView ORCID ID profile
https://doi.org/10.1186/s1366201819106
© The Author(s) 2018
 Received: 23 August 2018
 Accepted: 2 December 2018
 Published: 13 December 2018
Abstract
The construction of implicit Runge–Kutta–Nyström (RKN) method is considered in this paper. Based on the symmetric, symplectic, and exponentially fitted conditions, a class of implicit RKN integrators is obtained. The new integrators called ISSEFMRKN integrate exactly differential systems whose solutions are linear combinations of functions from the set \(\{\exp(\lambda t), \exp(\lambda t), \lambda\in\mathbb{C}\}\). In addition, their final stages also preserve the quadratic invariants \(\{\exp(2\lambda t), \exp(2\lambda t)\}\). Especially, we derived two methods: ISSEFMRKNs1o2 and ISSEFMRKNs2o4 which are of order 2 and 4, respectively. And the method ISSEFMRKNs2o4 has variable nodes. The derived method ISSEFMRKNs2o4 reduces to the classical RKN method (Qin and Zhu in Comput. Math. Appl. 22(9):85–95, 1991) as \(\lambda h\rightarrow0\). The numerical results show that our methods possess the efficiency and competence compared with some implicit RKN methods in the literature. Especially, ISSEFMRKNs2o4 improves the accuracy compared with unmodified method ISSEFRKNs2o4 proposed in (Zhai and Chen in Numer. Algebra Control Optim. 9(1):71–84, 2019).
Keywords
 Implicit
 Symmetric
 Symplectic
 Exponentially fitted
 Large step
MSC
 65L05
 65L06
 65M20
 65N40
 49M15
1 Introduction
As is known, the symplectic Runge–Kutta–Nyström methods share the pronounced property of being zerodissipative, which is an important requirement for solving oscillatory problem. Simos and VigoAguiar [26] then considered symplectic methods of RKN type which are adapted to certain types of oscillators. The term symplecticity essentially means area preserving in a phase space. Approximate solutions generated by symplectic methods are conservative even at finite resolution, in contrast with numerical methods that generate approximate solutions that are conservative only in the limit as the time step size approaches zero. Symplectic methods have been applied to many problems such as pendulum, Morse oscillator, harmonic oscillator, Lennard–Jones oscillator, Kepler’s orbit problem, and so on. In addition, it is pointed out in Chap. V and Chap. XI of [10] that symmetric methods show a better long time behavior than nonsymmetric ones when applied to reversible differential systems, as it is the case for conservative mechanical systems. So, some symmetric and symplectic RKN methods have been proposed such as [20].
However, they do not consider exponential fitting conditions. Exponentially fitted methods share very good behaviors when the solution of the problem can be expressed as linear combinations of functions from \(\{ \exp(\lambda t),\exp(\lambda t), \lambda\in\mathbb{C}\}\), or equivalently, from \(\{\sin(\omega t ),\cos(\omega t), \omega\in\mathbb{R}\}\) with \(\lambda =i\omega\), \(i^{2}=1\). The construction of exponentially fitted RK(N) methods is originally due to Paternoster [19]. After this, the exponentially fitted methods drew a lot of attention. As a result, there are many different types of exponentially fitted RK(N) methods such as [7, 8, 12, 16, 25, 26, 32].
In the last decade, much work related to symmetric, symplectic, and exponentially fitted conditions has emerged. These methods show a better behavior than the other methods which do not possess these three conditions together. The main work focuses on explicit methods for their easy implementation. However, the implicit methods are more suitable for solving stiff ODEs than the explicit methods. There are some researchers working on the implicit RKN methods, such as [13–15, 18]. Especially, in [33], the authors derived an implicit symmetric, symplectic, and exponentially fitted RKN integrator. Exactly, this method is not a true exponential fitting method. For the final stage coefficient \(b_{1}\), there are two different expressions. The authors chose \(\theta=\pm\frac{\sqrt{3}}{6}\) to make them as close as possible. In this paper, we want to avoid this from happening. So we investigate the modified Runge–Kutta–Nyström satisfying symmetric, symplectic, and EF conditions. Consequently, we can obtain unique expression of every coefficient which is not true for ISSEFRKN. The new method called ISSEFMRKN also reduces to the classical symplectic, symmetric RKN integrator when the parameter z approaches zero.
The remainder of the paper is organized as follows. We set up symmetric, symplectic, and EF conditions for our modified RKN methods in Sect. 2. In Sect. 3 we derive a class of twostage implicit symmetric symplectic exponentially fitted RKN (EFRKN) integrators. In Sect. 4 we present some numerical experiments that show the accuracy and efficiency of the new method when they are compared with other implicit RKN integrators given in [1, 20, 23, 24, 27, 33]. Finally, Sect. 5 is devoted to some conclusions.
2 Symmetric, symplectic, exponential fitting conditions
c  e  γ  A 
1  \(g_{2}\)  \(\bar{b}^{T}\)  
1  \(b^{T}\) 
\(c_{1}\)  1  \(\gamma_{1}\)  \(a_{11}\)  ⋯  \(a_{1s}\) 
⋮  ⋮  ⋮  ⋮  ⋱  ⋮ 
\(c_{s}\)  1  \(\gamma_{s}\)  \(a_{s1}\)  ⋯  \(a_{ss}\) 
1  \(g_{2}\)  \(\bar{b}_{1}\)  ⋯  \(\bar{b}_{s}\)  
1  \(b_{1}\)  ⋯  \(b_{s}\) 
2.1 Symmetric conditions
The concept of adjoint method is the hinge of symmetry. First, let us give the definition of adjoint method. We denote a onestep method for secondorder ODEs (1) as \(\varPhi_{h}: (y_{0},y'_{0})^{\mathrm{T}}\mapsto(y_{1},y'_{1})^{\mathrm{T}}\).
Definition 2.1
The adjoint method \(\varPhi_{h}^{*}\) of a onestep method \(\varPhi_{h}\) is the inverse map of the original method with reversed time step −h, i.e., \(\varPhi_{h}^{*}:=\varPhi_{h}^{1}\). In other words, \(y_{1}=\varPhi_{h}^{*}(y_{0})\) is implicitly defined by \(\varPhi_{h}(y_{1})=y_{0}\). A method for which \(\varPhi_{h}^{*}=\varPhi_{h}\) is called symmetric (see [11]).
2.2 Symplectic conditions
Definition 2.2
2.3 Exponential fitting conditions

for the internal stages,$$ \varphi_{i} \bigl[y(t);h;{\mathbf {a}} \bigr]=y(t+c_{i}h)y(t)c_{i} \gamma_{i}hy'(t)h^{2}\sum _{j=1}^{s}a_{ij}y''(t+c_{j}h), \quad i=1,2,\ldots,s; $$

for the final stages,$$ \begin{aligned} &\varphi \bigl[y(t);h;{\bar{\mathbf {b}}} \bigr]=y(t+h)y(t)hg_{2}y'(t)h^{2} \sum _{i=1}^{s}\bar{b}_{i}y''(t+c_{i}h), \\ &\varphi \bigl[y(x);h;{\mathbf {b}} \bigr]=y'(t+h)y'(t)h \sum_{i=1}^{s}b_{i}y''(t+c_{i}h). \end{aligned} $$
2.4 Algebraic order conditions
3 Construction of implicit symmetric and symplectic EFRKN methods
In this section we construct an implicit EFRKN method under the symmetry, symplecticity, exponential fitting conditions obtained in the previous section.

\(s=1\)

\(s=2\)
4 Numerical experiments

DIRKN4(3): The embedded diagonally implicit RKN 4(3) pair method proposed by AlKhasawneh et al. in [1].

DIRKNs3o4: The diagonally implicit threestage RKN of order four proposed by Senu et al. in [23].

ISSRKNs2o4: The implicit symmetric and symplectic twostage RKN of order four proposed by Qin et al. in [20] with \(a_{11}=\frac{1}{72}\).

ISSEFRKN2: The implicit symmetric and symplectic exponentially fitted twostage RKN of order four proposed in [33].

ISSEFMRKNs1o2: The implicit symmetric and symplectic exponentially fitted onestage modified RKN (11) of order two proposed in this paper.

ISSEFMRKNs2o4: The implicit symmetric and symplectic exponentially fitted twostage modified RKN (23) of order four proposed in this paper.
Problem 1
Problem 2
In this experiment, we choose \(e = 10^{2}\). Figure 2(b) and (d) present the efficiency curves for the Hamiltonian and for the momentum, respectively, on the interval \([0, 500]\) with the step sizes \(h = 1/2^{m}\), \(m=1, \ldots, 4\).
Problem 3
From Figs. 1–3, we can find that the implicit modified EFRKN method ISSEFMRKNs2o4 is more efficient than other methods used for comparison in this paper. However, sometimes the method ISSEFMRKNs1o2 performs not better than other methods. From Fig. 2 and Fig. 3, we can see that symplectic methods ISSRKNs2o4, ISSEFRKNs2o4, ISSEFMRKNs1o2, and ISSEFMRKNs2o4 show a better behavior than unsymplectic methods DIRKNs3o4 and DIRKN4(3). This illustrates the special property – symplecticity. Especially, for problem 3, symplectic methods preserve the Hamiltonian much more accurately than the methods which are not symplectic. Comparing the modified method ISSEFMRKNs2o4 and the unmodified code ISSEFRKNs2o4 proposed in [33], we find that ISSEFMRKNs2o4 gives a better behavior than ISSEFRKNs2o4, although they have the same order (order 4).
5 Conclusions
In this paper, we focus on a class of twostage IEFMRKN integrators which are symmetric and symplectic and exponentially fitted. In addition, their final stages also preserve the quadratic invariants \(\{\exp(2\lambda t), \exp(2\lambda t)\}\). Like some existing EFRKN integrators (see [17, 32, 33] for example), the coefficients of the new methods depend on the product of the dominant frequency ω and the step size h. Especially, we derived two ISSEFMRKN methods: ISSEFMRKNs1o2 and ISSEFMRKNs2o4. They reduce to the classical RKN method as the parameter λh approaches zero. From the numerical experiments, we can find that these two new methods are more efficient than unmodified ISSEFRKN and some other integrators used for comparison. However, the algebraic order of ISSEFMRKNs1o2 and ISSEFMRKNs2o4 are 2, 4, respectively, which is not high. Thus in the future, we will consider the derivation of high order ISSEFRKN methods, such as six order, eight order.
Declarations
Acknowledgements
The authors are grateful to the anonymous referees for their careful reading of the manuscript and for their invaluable comments and suggestions, which largely helped to improve this paper.
Availability of data and materials
No applicable.
Funding
The research of Wenjuan Zhai was supported in part by the Project for Youth Scholars of Higher Education of Hebei Province (QN2017402), and the Project of Teaching and Research of Beijing Jiaotong University Haibin College (HBJY16005).
Authors’ contributions
HZ derived the method ISSEFMRKNs1o2. And he redid all the new numerical experiments. He also refined the English of our manuscript. WZ derived the symmetric conditions, symplectic conditions, and exponentially fitted conditions and gave the method ISSEFMRKNs2o4. BC wrote the first draft and organized the whole paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 AlKhasawneh, R.A., Ismail, F., Suleiman, M.: Embedded diagonally implicit Runge–Kutta–Nyström 4(3) pair for solving special secondorder IVPs. Appl. Math. Comput. 190, 1803–1814 (2007) MathSciNetMATHGoogle Scholar
 Albrecht, P.: The extension of the theory of Amethods to RK methods. In: Strehmel, K. (ed.) Numerical Treatment of Differential Equations. Proceedings of the 4th Seminar NUMDIFF4, TuebnerTexte Zur Mathematik, pp. 8–18. Tuebner, Leipzig (1987) Google Scholar
 Albrecht, P.: A new theoretical approach to Runge–Kutta methods. SIAM J. Numer. Anal. 24, 391–406 (1987) MathSciNetMATHView ArticleGoogle Scholar
 Calvo, M.P., SanzSerna, J.M.: Highorder symplectic Runge–Kutta–Nyström methods. SIAM J. Sci. Comput. 14, 1237–1252 (1993) MathSciNetMATHView ArticleGoogle Scholar
 Coleman, J.P., Ixaru, L.Gr.: Pstability and exponentialfitting methods for \(y''=f(x,y)\). IMA J. Numer. Anal. 6, 179–199 (1996) MathSciNetMATHGoogle Scholar
 Franco, J.M.: RungeKuttaNyström methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Commun. 147, 770–782 (2002) MATHView ArticleGoogle Scholar
 Franco, J.M.: Exponentially fitted explicit Runge–Kutta–Nyström methods. J. Comput. Appl. Math. 167, 1–19 (2004) MathSciNetMATHView ArticleGoogle Scholar
 Franco, J.M.: Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems. Comput. Phys. Commun. 177, 479–492 (2007) MathSciNetMATHView ArticleGoogle Scholar
 Franco, J.M., Gomez, I.: Symplectic explicit methods of Runge–Kutta–Nyström type for solving perturbed oscillators. J. Comput. Appl. Math. 260, 482–493 (2014) MathSciNetMATHView ArticleGoogle Scholar
 Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2002) MATHView ArticleGoogle Scholar
 Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I, Nonstiff Problems, Second revised edn. Springer, Berlin (1993) MATHGoogle Scholar
 Ixaru, Gr.L., Vanden Berghe, G.: Exponential Fitting. Kluwer Academic Publishers, Dordrecht (2004) MATHView ArticleGoogle Scholar
 Jator, S.N.: Implicit third derivative Runge–Kutta–Nyström method with trigonometric coefficients. Numer. Algorithms 70(1), 1–18 (2015) MathSciNetMATHView ArticleGoogle Scholar
 Kalogiratou, Z.: Diagonally implicit trigonometrically fitted symplectic Runge–Kutta methods. Appl. Math. Comput. 219(14), 7406–7412 (2013) MathSciNetMATHGoogle Scholar
 Kalogiratou, Z., Monovasilis, T., Simos, T.E.: A sixth order symmetric and symplectic diagonally implicit Runge–Kutta method. In: International Conference of Computational Metho. vol. 1618, pp. 833–838. Am. Inst. of Phys., New York (2014) Google Scholar
 Li, J., Deng, S., Wang, X.: Extended explicit pseudo twostep RKN methods for oscillatory systems \(y'' + My = f(y)\). Numer. Algorithms 78, 673–700 (2018) MathSciNetMATHGoogle Scholar
 Li, J., Wang, X., Deng, S., Wang, B.: Trigonometricallyfitted symmetric twostep hybrid methods for oscillatory problems. J. Comput. Appl. Math. 344, 115–131 (2018) MathSciNetMATHView ArticleGoogle Scholar
 Moo, K.W., Senu, N., Ismail, F., Arifin, N.M.: A zerodissipative phasefitted fourth order diagonally implicit Runge–Kutta–Nyström method for solving oscillatory problems. Math. Probl. Eng. 2, 1–8 (2014) View ArticleGoogle Scholar
 Paternoster, B.: Runge–Kutta(–Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials. Appl. Numer. Math. 28, 401–412 (1998) MathSciNetMATHView ArticleGoogle Scholar
 Qin, M.Z., Zhu, W.J.: Canonical Runge–Kutta–Nyström methods for second order ordinary differential equations. Comput. Math. Appl. 22(9), 85–95 (1991) MathSciNetMATHView ArticleGoogle Scholar
 SanzSerna, J.M.: Symplectic integrators for Hamiltonian problems: an overview. Acta Numer. 1, 243–286 (1992) MathSciNetMATHView ArticleGoogle Scholar
 SanzSerna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994) MATHView ArticleGoogle Scholar
 Senu, N., Suleiman, M., Ismail, F., Othman, M.: A new diagonally implicit Runge–Kutta–Nyström method for periodic IVPs. WSEAS Trans. Math. 9(9), 679–688 (2010) MathSciNetMATHGoogle Scholar
 Sharp, P.W., Fine, J.M., Burrage, K.: Two stage and three stage diagonally implicit Runge–Nutta–Nyström methods of orders three and four. IMA J. Numer. Anal. 10, 489–504 (1990) MathSciNetMATHView ArticleGoogle Scholar
 Simos, T.E.: An exponentiallyfitted Runge–Kutta method for the numerical integration of initialvalue problems with periodic or oscillating solutions. Comput. Phys. Commun. 115, 1–8 (1998) MathSciNetMATHView ArticleGoogle Scholar
 Simos, T.E., VigoAguiar, J.: Exponentially fitted symplectic integrator. Phys. Rev. E 67, 1 (2003) MathSciNetView ArticleGoogle Scholar
 Van der Houwen, P.J., Sommeijer, B.P.: Diagonally implicit Runge–Nutta–Nyström methods for oscillating problems. SIAM J. Numer. Anal. 26(2), 414–429 (1989) MathSciNetMATHView ArticleGoogle Scholar
 Vanden Berghe, G., De Meyer, H., Van Daele, M., Van Hecke, T.: Exponentially fitted Runge–Kutta methods. J. Comput. Appl. Math. 125, 107–115 (2000) MathSciNetMATHView ArticleGoogle Scholar
 Vanden Berghe, G., Van Daele, M., Van de Vyver, H.: Exponential fitted Runge–Kutta methods of collocation type: fixed or variable knot points? J. Comput. Appl. Math. 159, 217–239 (2003) MathSciNetMATHView ArticleGoogle Scholar
 Wang, B., Meng, F., Fang, Y.: Efficient implementation of RKN type Fourier collocation methods for secondorder differential equations. Appl. Numer. Math. 119, 164–178 (2017) MathSciNetMATHView ArticleGoogle Scholar
 Wang, B., Yang, H., Meng, F.: Sixth order symplectic and symmetric explicit ERKN schemes for solving multifrequency oscillatory nonlinear Hamiltonian equations. Calcolo 54, 117–140 (2017) MathSciNetMATHView ArticleGoogle Scholar
 You, X., Chen, B.: Symmetric and symplectic exponentially fitted Runge–Kutta(–Nyström) methods for Hamiltonian problems. Math. Comput. Simul. 94, 76–95 (2013) View ArticleGoogle Scholar
 Zhai, W., Chen, B.: A fourth order implicit symmetric and symplectic exponentially fitted Runge–Kutta–Nyström method for solving oscillatory problems. Numer. Algebra Control Optim. 9(1), 71–84 (2019) View ArticleGoogle Scholar